RESEARCH ARTICLE

von Neumann’s mean ergodic theorem on complete random inner product modules

  • Xia ZHANG ,
  • Tiexin GUO
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  • LMIB and School of Mathematics and Systems Science, Beihang University, Beijing 100191, China

Received date: 29 Jan 2011

Accepted date: 14 Apr 2011

Published date: 01 Oct 2011

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on Lp(,H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, Lp(,H) (1≤p<∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, , P), a sub σ-algebra of , and Lp(,H) the complete random normed module generated by Lp(,H).

Cite this article

Xia ZHANG , Tiexin GUO . von Neumann’s mean ergodic theorem on complete random inner product modules[J]. Frontiers of Mathematics in China, 2011 , 6(5) : 965 -985 . DOI: 10.1007/s11464-011-0139-4

1
Albanese A A, Bonet J, Ricker W J. C0-semigroups and mean ergodic operators in a class of Fréchet spaces. J Math Anal Appl, 2010, 365: 142-157

DOI

2
Beck A, Schwartz J T. A vector-valued random ergodic theorem. Proc Am Math Soc, 1957, 8: 1049-1059

DOI

3
Chen P D. The theory of random measures. Acta Math Sinica, 1976, 19: 210-216 (in Chinese)

4
Dunford N, Schwartz J T. Linear Operators (I). New York: Interscience, 1957

5
Filipović D, Kupper M, Vogelpoth N. Separation and duality in locally L0-convex modules. J Funct Anal, 2009, 256: 3996-4029

DOI

6
Fonf V P, Lin M, Wojtaszczyk P. Ergodic characterizations of reflexivity of Banach spaces. J Funct Anal, 2001, 187: 146-162

DOI

7
Guo T X. Extension theorems of continuous random linear operators on random domains. J Math Anal Appl, 1995, 193: 15-27

DOI

8
Guo T X. The Radon-Nikodým property of conjugate spaces and the w-equivalence theorem for w-measurable functions. Sci China Ser A, 1996, 39: 1034-1041

9
Guo T X. Module homomorphisms on random normed modules. Chinese Northeastern Math J, 1996, 12: 102-114

10
Guo T X. Some basic theories of random normed linear spaces and random inner product spaces. Acta Anal Funct Appl, 1999, 1: 160-184

11
Guo T X. The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure. Sci China Ser A, 2008, 51: 1651-1663

DOI

12
Guo T X. Relations between some basic results derived from two kinds of topologies for a random locally convex module. J Funct Anal, 2010, 258: 3024-3047

DOI

13
Guo T X. Recent progress in random metric theory and its applications to conditional risk measures. Sci China Ser A, 2011, 54(4): 633-660

DOI

14
Guo T X. The theory of module homomorphisms in complete random inner product modules and its applications to Skorohod’s random operator theory. Preprint

15
Guo T X, Li S B. The James theorem in complete random normed modules. J Math Anal Appl, 2005, 308: 257-265

DOI

16
Guo T X, Shi G. The algebraic structure of finitely generated L0(ℱ,K)-mod⁡ules and the Helly theorem in random normed modules. J Math Anal Appl, 2011, 381: 833-842

DOI

17
Guo T X, Xiao H X, Chen X X. A basic strict separation theorem in random locally convex modules. Nolinear Anal: TMA, 2009, 71: 3794-3804

DOI

18
Guo T X, You Z Y. The Riesz’s representation theorem in complete random inner product modules and its applications. Chin Ann of Math, Ser A, 1996, 17: 361-364 (in Chinese)

19
Guo T X, Zeng X L. Random strict convexity and random uniform convexity in random normed modules. Nonlinear Anal, 2010, 73: 1239-1263

DOI

20
Guo T X, Zhang X. Stone’s representation theorem on complete complex random inner product modules. Preprint

21
Petersen K. Ergodic Theory. Cambridge Studies in Advanced Mathematics 2. London-New York-New Rochelle-Melbourne-Sydney: Cambridge University Press, 1983

22
von Neumann J. Proof of the quasi-ergodic hypothesis. Proc Nat Acad Sci, 1932, 18: 70-82

DOI

23
Yosida K, Kakutani S. Operator theoretical treatment of Markoff’s process and the mean ergodic theorem. Ann of Math, 1941, 42: 188-228

DOI

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